Actually, I noticed that literally only 1 L-shaped x-y-x cage wasn't eliminated purely by the single digit cages, 5-1-5 in the 11+.

So now I'm wondering if it's statistically more probably that an L-shaped x-y-x occurs, or if we just hit a really weird random stretch.

Yes, I also have that feeling, but to calculate the probability is really difficult, requiring a complex calculation. Instead, we could approach it by obtaining the "frequency", but we would need the software to create randomly an enough wide sample, let's say, 10,000 (or 100,000) of those puzzles verifying the unicitiy of the solution in all cases of course, and counting how many of them have the L-shape x-y-x referred.

(I attach today the yesterday's official solution for future reference):

sneaklyfox

Posted on:Thu Feb 09, 2012 6:19 pm

Posts: 422Location: CanadaJoined: Fri May 13, 2011 2:43 am

Re: Something I've always been curious about

Well, there is still an x-y-x cage in b1-c1-c2. All the other puzzles could be like that if you just flip them...

Well, there is still an x-y-x cage in b1-c1-c2. All the other puzzles could be like that if you just flip them...

No, no, sneaklyfox, that cage has nothing to do, if you make a central symmetry (or a rotation 180º degrees right), moving the upper left box to the bottom right box you obtain a totally different configuration but not this one. This was discussed in the past in this same thread, I reproduce the complete paragraph I included in my analysis on Nov 21, 2011:

"Usually the top left area is solved first and quickly and the configuration in this area has "nothing to do" with the bottom right area since the puzzle is not symmetric (keeping the same aspect) in the sense that it would not be valid to rotate, for instance, 180 degrees to the right because we would obtain a different puzzle (only the cages themselves are symmetric, i.e., inside the blue areas, but not the full puzzle), we may observe that all six 3-cell in-line cages “look inside” the top left area providing, in some way, the required information, while what we could say of the bottom right area is that it is “blind” in this context. Then we will forget the symmetries of the full puzzle and work with the original configuration, the one of the graphic."

sneaklyfox

Posted on:Thu Feb 09, 2012 8:58 pm

Posts: 422Location: CanadaJoined: Fri May 13, 2011 2:43 am

Re: Something I've always been curious about

clm wrote:

sneaklyfox wrote:

Well, there is still an x-y-x cage in b1-c1-c2. All the other puzzles could be like that if you just flip them...

No, no, sneaklyfox, that cage has nothing to do, if you make a central symmetry (or a rotation 180º degrees right), moving the upper left box to the bottom right box you obtain a totally different configuration but not this one. This was discussed in the past in this same thread, I reproduce the complete paragraph I included in my analysis on Nov 21, 2011:

"Usually the top left area is solved first and quickly and the configuration in this area has "nothing to do" with the bottom right area since the puzzle is not symmetric (keeping the same aspect) in the sense that it would not be valid to rotate, for instance, 180 degrees to the right because we would obtain a different puzzle (only the cages themselves are symmetric, i.e., inside the blue areas, but not the full puzzle), we may observe that all six 3-cell in-line cages “look inside” the top left area providing, in some way, the required information, while what we could say of the bottom right area is that it is “blind” in this context. Then we will forget the symmetries of the full puzzle and work with the original configuration, the one of the graphic."

Sorry if I misunderstood. I guess I have not been reading in all the detail about this thread, but just thought that the discussion was based on the appearance of an x-y-x L-shaped cage which seemed to occur in every puzzle. Although the configuration is not the same because the 3-cell in-line cages would be oriented differently, I don't think that fact affects so very much. Perhaps every puzzle of this configuration has an x-y-x L-shaped cage somewhere whether it's in the "blind" bottom right area or the top left area.

... ... Sorry if I misunderstood. I guess I have not been reading in all the detail about this thread, but just thought that the discussion was based on the appearance of an x-y-x L-shaped cage which seemed to occur in every puzzle. Although the configuration is not the same because the 3-cell in-line cages would be oriented differently, I don't think that fact affects so very much. Perhaps every puzzle of this configuration has an x-y-x L-shaped cage somewhere whether it's in the "blind" bottom right area or the top left area.

Yes, that was the original discussion, with the x-y-x L-shape cage in the right bottom area (3x3 box). With respect to your new hypothesis, I am affraid not. Once we have a puzzle that breaks the original hypothesis discussed in this thread it is not very difficult to obtain a variation like this (I have only permuted the numbers 2, 3, 4 and 6, in the first two rows, and consequently modify the result of four cages):

and, if you are so kind to confirm the unicity of the solution (the numbers in blue are different to the yesterday's solution, the numbers in black are the same), here we can see an exception where no cage is of the type x-y-x:

starling

Posted on:Fri Feb 10, 2012 1:03 am

Posts: 175Joined: Fri May 13, 2011 2:11 am

Re: Something I've always been curious about

That said, I wonder if sneaklyfox is right that this puzzle broke that rule because of the other x-y-x L shaped cage. Maybe it's just that there has to be one.

That said, I wonder if sneaklyfox is right that this puzzle broke that rule because of the other x-y-x L shaped cage. Maybe it's just that there has to be one.

But, since there is at least one puzzle (the one I provided yesterday, in my last reply to sneaklyfox, I believe with an unique solution) with that type of configuration and without any 3-cell L-shape cages of the type x-y-x, we must then accept the possbility of finding a big amount of them.